# Thread: Evaluate the integral of 2cos(lnt)dt

1. ## Evaluate the integral of 2cos(lnt)dt

by using substitution and then separation of variables.

i'm lost.

2. Originally Posted by -DQ-
by using substitution and then separation of variables.

i'm lost.
Substitute $z := \ln t$, and, therefore $dt = t\cdot dz=e^z\, dz$, to get

$\int 2\cos(\ln t)\, dt=\int 2\cos(z)\cdot e^z\,dz=e^z\cdot \left(\cos(z)+\sin(z)\right)+C$
$=t\cdot\left(\cos(\ln t)+\sin(\ln t)\right)+C$
To determine the value of $\int 2\cos(z)\cdot e^z\,dz$, you do partial integration twice, and then solve for $\int 2\cos(z)\cdot e^z\,dz$.